Real Numbers Class 10: Important Questions & Worksheet PDF
Practise prime factorisation, HCF, LCM and irrationality with 10 original Class 10 questions, worked answers and a free printable worksheet.
Use this 30-minute worksheet after studying prime factors. Questions 1–5 check calculation; questions 6–10 ask you to choose a method or justify a claim. These are original revision questions, not predictions of board-exam questions. Write your working before looking at the answers.
Real numbers practice questions
- 1. Express 180 and 252 as products of primes.
- 2. Find the HCF and LCM of 180 and 252.
- 3. A teacher has 84 pencils and 126 erasers. Make the greatest possible number of identical packs with no items left. How many packs and what goes in each?
- 4. Two bells ring together at 9:00 a.m. and repeat every 12 and 18 minutes. When will they next ring together?
- 5. Find the least positive integer divisible by 8, 12 and 15.
- 6. The HCF of two positive integers is 6 and their LCM is 180. One number is 30. Find the other, then verify the claim.
- 7. Can a positive integer power of 3 end in zero? Explain using prime factors.
- 8. Prove that the square root of 7 is irrational.
- 9. If a number r is irrational, must r + 4 be irrational? Give a proof.
- 10. Does HCF × LCM equal the product for three numbers? Test 2, 4 and 8.
Worked answers and method
1–2. Prime factors, HCF and LCM
- 1180 = 2² × 3² × 5; 252 = 2² × 3² × 7.
- 2The shared prime powers with the smaller exponents give HCF = 2² × 3² = 36.
- 3Use the largest exponent of every prime for LCM = 2² × 3² × 5 × 7 = 1260.
- 4Check: 36 × 1260 = 45360 = 180 × 252.
3. Identical packs
- 1The number of packs must divide both 84 and 126. The greatest such divisor is HCF(84,126) = 42.
- 2Each pack has 84 ÷ 42 = 2 pencils and 126 ÷ 42 = 3 erasers. Packs = 42, not 42 items per pack.
4–5. Repeating events and common multiples
- 1LCM(12,18) = 36, so the bells ring together at 9:36 a.m.
- 28 = 2³, 12 = 2² × 3, 15 = 3 × 5. Their LCM is 2³ × 3 × 5 = 120.
6. Use the product identity and check
- 1The other number is (6 × 180) ÷ 30 = 36.
- 2HCF(30,36) = 6 and LCM(30,36) = 180, so both conditions hold.
7. Last digit zero
- 1A number ending in zero is divisible by 10 and therefore has prime factors 2 and 5.
- 2A positive integer power of 3 has only the prime factor 3, so it cannot end in zero.
8. Irrationality proof
- 1Assume √7 = a/b in lowest terms, where a and b are integers and b ≠ 0. Then a² = 7b².
- 2Because 7 is prime, 7 dividing a² means 7 divides a. Write a = 7k.
- 3Substitution gives b² = 7k², so 7 also divides b. This contradicts a/b being in lowest terms. Therefore √7 is irrational.
9–10. Reasoning checks
- 1If r + 4 were rational, subtracting rational 4 would make r rational, a contradiction.
- 2For 2, 4 and 8, HCF = 2 and LCM = 8. Their product is 16, while 2 × 4 × 8 = 64. The two-number identity does not extend unchanged.
How to revise mistakes
If you chose LCM for the packing problem, underline “greatest number of identical packs”: the number must divide both totals. For bells, list the first few multiples to see why the first shared time uses LCM. For a proof, check that you explicitly state the fraction is in lowest terms before reaching a contradiction.